I believe - off-hand, and with no evidence - that the American way of writing dates is in fact more old-fashioned. If you look at old books - from the nineteenth or early twentieth century - they tend to write dates American-style. I suspect that what's happened is that everyone else has changed to the more logical style and the Americans have stayed where they are - just like metric/imperial.
It's a little like -ise and -ize for words like "customise/customize". Either spelling is acceptable in English no matter where you are, but "-ize" is more common in America whilst "-ise" is usually used elsewhere. If you look at those old books again you'll find that "-ize" is much more common. I think that it's the Americans who have been conservative whilst language has become more logical elsewhere. "-ise" is more logical, you see, because if you use the "-ize" system there are some words that still always take "-ise" - such as "circumcise". You can never write "circumsize" even if you're using "-ize" normally. So it's simpler just to "-ise" everything and not have to worry.
It's a little like -ise and -ize for words like "customise/customize". Either spelling is acceptable in English no matter where you are, but "-ize" is more common in America whilst "-ise" is usually used elsewhere. If you look at those old books again you'll find that "-ize" is much more common. I think that it's the Americans who have been conservative whilst language has become more logical elsewhere. "-ise" is more logical, you see, because if you use the "-ize" system there are some words that still always take "-ise" - such as "circumcise". You can never write "circumsize" even if you're using "-ize" normally. So it's simpler just to "-ise" everything and not have to worry.